MathDB

2007 Iran MO (3rd Round)

Part of Iran MO (3rd Round)

Subcontests

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New numbers between real numbers

Scientist have succeeded to find new numbers between real numbers with strong microscopes. Now real numbers are extended in a new larger system we have an order on it (which if induces normal order on R \mathbb R), and also 4 operations addition, multiplication,... and these operation have all properties the same as R \mathbb R. http://i14.tinypic.com/4tk6mnr.png a) Prove that in this larger system there is a number which is smaller than each positive integer and is larger than zero. b) Prove that none of these numbers are root of a polynomial in R[x] \mathbb R[x].

Irreducible

Something related to this [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=845756#845756]problem: Prove that for a set S⊂N S\subset\mathbb N, there exists a sequence \{a_{i}\}_{i \equal{} 0}^{\infty} in S S such that for each n n, \sum_{i \equal{} 0}^{n}a_{i}x^{i} is irreducible in Z[x] \mathbb Z[x] if and only if ∣S∣≥2 |S|\geq2. By Omid Hatami
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